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# StretchedExpFT¶

## Properties (fitting parameters)¶

Name

Default

Description

Height

0.1

Intensity at the origin

Tau

100.0

Relaxation time

Beta

1.0

Stretching exponent

Centre

0.0

Centre of the peak

## Description¶

Provides the Fourier Transform of the Symmetrized Stretched Exponential Function

$S(Q,E) = Height \int_{-\infty}^{\infty} dt/h \cdot e^{-i2\pi (E-Centre)t/h} \cdot e^{-|\frac{t}{Tau}|^{Beta}} )$

with $$h$$ Planck’s constant. If the energy units of energy are micro-eV, then tau is expressed in pico-seconds. If E-units are micro-eV then tau is expressed in nano-seconds.

Properties:

• Normalization $$\int_{-\infty}^{\infty} dE \cdot S(Q,E) = Height$$

• Maximum $$S(Q,E\equiv 0)=Height \cdot Tau \cdot Beta^{-1} \cdot \Gamma(Beta^{-1})$$

## Usage¶

Note

Example - Fit to a QENS signal:

The QENS signal is modeled by the convolution of a resolution function with elastic and StretchedExpFT components. Noise is modeled by a linear background:

$$S(Q,E) = R(Q,E) \otimes (\alpha \delta(E) + StretchedExpFT(Q,E)) + (a+bE)$$

Obtaining an initial guess close to the optimal fit is critical. For this model, it is recommended to follow these steps: - In the Fit Function window of a plot in MantidWorkbench, construct the model. - Tie parameter $$Height$$ of StretchedExpFT to zero, then carry out the Fit. This will result in optimized elastic line and background. - Untie parameter $$Height$$ of StretchedExpFT and tie parameter $$Beta$$ to 1.0, then carry out the fit. This will result in optimized model using an exponential. - Release the tie on Beta and redo the fit.

# Load resolution function and scattered signal

# This function_string is obtained by constructing the model
# with the Fit Function window of a plot in MantidWorkbench, then
# Setup--> Manage Setup --> Copy to Clipboard
function_string  = "(composite=Convolution,FixResolution=true,NumDeriv=true;"
function_string += "name=TabulatedFunction,Workspace=resolution,WorkspaceIndex=0,Scaling=1,Shift=0,XScaling=1;"
function_string += "(name=DeltaFunction,Height=1,Centre=0;"
function_string += "name=StretchedExpFT,Height=1.0,Tau=100,Beta=0.98,Centre=0));"
function_string += "name=LinearBackground,A0=0,A1=0"

# Carry out the fit. Produces workspaces  fit_results_Parameters,
#  fit_results_Workspace, and fit_results_NormalisedCovarianceMatrix.
Fit(Function=function_string,
InputWorkspace="qens_data",
WorkspaceIndex=0,
StartX=-0.15, EndX=0.15,
CreateOutput=1,
Output="fit_results")

# Collect and print parameters for StrechtedExpFT
parameters_of_interest = ("Tau", "Beta")
values_found = {}
ws = mtd["fit_results_Parameters"]  # Workspace containing optimized parameters
for row_index in range(ws.rowCount()):
full_parameter_name = ws.row(row_index)["Name"]
for parameter in parameters_of_interest:
if parameter in full_parameter_name:
values_found[parameter] = ws.row(row_index)["Value"]
break
if values_found["Beta"] > 0.63 and values_found["Beta"] < 0.71:
print("Beta found within [0.63, 0.71]")
if values_found["Tau"] > 54.0 and values_found["Tau"] < 60.0:
print("Tau found within [54.0, 60.0]")


Output:

Beta found within [0.63, 0.71]
Tau found within [54.0, 60.0]


Categories: FitFunctions | QuasiElastic

## Source¶

Python: StretchedExpFT.py